where \(s_j\)  the dominant eigenvector of \(M_{ij}\)  with largest (least negative) eigenvalue \(-q\)  [1]}, defined in Sec. REF .
With (REF ) and (REF ), the prior predictive distribution (REF ) gives
\(\boxed{P(I|{\cal H}_{\rm late}) \simeq \frac{ \sum \limits _{j} \kappa _{Ij} s_j \int \limits _0^{t_{\rm c}} {\rm d} t\, a^3(t) {\rm e}^{-q t}}{\sum \limits _{k} \kappa _{k}s_k \int \limits _0^{t_{\rm c}} {\rm d} t \, a^3(t) {\rm e}^{-q t}} = \frac{\sum \limits _j \kappa _{Ij} s_j }{\sum \limits _{k} \kappa _{k} s_k} }\,.\) 
